No a bad convrter can not damage your transmission however it will make your vehicale run rough
Transmission torque converter or an exhuast catalytic converter?
TCC (Torque converter clutch) Soleniod in the transmission is likely bad. Hope for that because if not it's a bad torque converter.
Slippage in the transmission or a bad torque converter
If its an automatic I would bet money its the torque converter clutch soloniod in the valve body in the transmission, if its not that its probably a bad torque converter.
It's a good idea.
Parking brake not released, bad torque converter, bad transmission shift solenoid or bad transmission in general, broken shift cable not engaging the transmission.
Low or bad transmission oil, bad transmission solenoid, vacuum line issues, worn transmission gears, bad torque converter.. all possible reasons.
Loose timing chain? Loose heat shield on catalytic converter?
Torque converter going bad
Bad torque converter would be a likely reason, especially with the 4L60E
Sounds like torque converter is going bad
Bad torque converter would be my first suspect. Bad vacuum pump is another possibility.